Resignation has arrived at Queens Park Rangers. The approach of relegation was not always evident in their performance at Everton but it was clear in song. "Next year, we'll be champions" chanted the small pocket of away fans and they were not referring to their current division.

Everton comfortably won a game that had remained in the balance for only 40 minutes. The catalyst was the reaction of Rangers' players to a deflected Darron Gibson opener that paved the way for a dominant second-half display from David Moyes' European hopefuls and a deserved goal for the impressive Victor Anichebe. "Our heads dropped," Harry Redknapp said. "Once we went 1-0 down we lost belief."

The QPR manager would not accept that relegation is inevitable – "What can I say? We have to keep going," he argued – but he knows. Seven points from safety is surmountable but they lack the games, only five, the defensive discipline and belief to conjure a remarkable reprieve. By contrast Everton are heading into a key game at Arsenal on Tuesday with renewed resolve.

"It was always going to be difficult," said Redknapp in a telling past tense. "A third of the season was gone and we had only four points so it was never going to be easy. You take a team on at that stage that isn't yours and try to get the best out of them. But it was always going to be difficult."

The opportunity on offer at Goodison added to the torment for the visiting manager. QPR showed no immediate effect from the stoppage-time Wigan equaliser last weekend that Redknapp labelled "the cruellest last kick" of his 30 years of management. Following an immaculately observed minute's silence for the victims of Hillsborough before the 24th anniversary of the disaster, the visitors initially stood firm against a powerful Everton attack and offered a potent threat of their own. The pace of Andros Townsend was a productive outlet on the right wing, Junior Hoilett almost profited from their first attack and it took a fine save from Tim Howard to deny Loïc Rémy.

Rangers were vibrant on the counter-attack but their defensive resilience was stretched by the strength of Anichebe plus the improving relationship between Kevin Mirallas and Seamus Coleman on the Everton right. José Bosingwa endured a torrid afternoon against the Republic of Ireland defender, a 34th-minute booking for persistent misconduct encapsulating his hapless efforts.

Everton looked to hit Anichebe early and prosper from the Nigerian's impressive first touch. The striker could have opened the scoring from a Leighton Baines cross, struck powerfully at close range admittedly, but steered a diving header across Júlio César's goal. Sylvain Distin squandered a clear opening when Gibson's low drive landed at his feet seven yards from goal.

The home side sustained the pressure and broke Rangers' resistance with a goal that fulfilled the cliche of luck deserting a struggler. Everton worked an attack out to Gibson who let fly from over 25 yards. Júlio César had the shot covered only to be completely wrongfooted as it struck the inside of Clint Hill's leg and deflected into the opposite corner.

The breakthrough, and the manner of it, visibly drained QPR's confidence and desire. Everton could easily have scored another three before Anichebe doubled their advantage. QPR did not learn the lesson from an earlier Baines corner towards Distin at the far post, denied the first time by Júlio César, and when the routine was repeated the defender headed down for Anichebe to sweep in from close range.

"I remember bringing Victor on as a substitute and hearing the groans coming down from the stands," said Moyes, often alone in his faith in the forward. "But he has got all the attributes. He is a terrific player and has got all the bits that top centre-forwards have. Some of his hold-up play was terrific. The big change in him has been self-confidence."